Extra Credit Assignment #2
Fall 2011, Math 1210-007
Name:
Instructions: You may submit your solutions to this assignment at any time BEFORE 10:45 a.m., Friday, November 4, 2011. Read each problem carefully, and follow the directions. Write neatly and clearly. Show all of your work. No partial credit will be awarded. Include your name on each sheet you submit, including any computer printouts. Number each page of your assignment. Staple all pages, including this cover sheet , together.
This assignment is out of 100 points total.
Let f ( x ) = 2 x − cos (4 x ) on the interval I = [ − π, π ] .
1.
(5 points) Find all critical points of f on I .
2.
(2 points) Where is f increasing and where is it decreasing?
3.
(2 points) Where do local maxima and minima occur?
4.
(2 points) What are the global maximum and global minimum values of f ?
5.
(7 points) Where is f concave up and concave down?
6.
(2 points) Name all points of inflection that have pairs.
x -coordinates in I . Your answer should be in ordered
7.
(5 points) Using the information above, sketch a graph of your graph.
8.
(5 points) Find the linear approximation L ( x ) to f at a =
π
8
.
f by hand. Label each inflection point on
9.
(5 points) Sketch f ( x ) and L ( x ) on the same set of axes over the interval 0 ,
π
4
.
*For the following problems, you will need to use a computer (time saver) or calculator (time sink). If you’ve never used the computer to do your bidding before, now is a great time to learn. For non-programmers, I would suggest Maple. You have access to several programs in the computer lab located in the Math Center (on the other side of the tutoring center). The computer lab tech can help you with logging on to the math department computers and opening applications, but his/her job is not to do your assignment for you. Be proactive and self-sufficient; it’s the best way to learn! Remember, the help menu and Google (in moderation) are your friends.
10.
(15 points) Use the bisection method to approximate the solution to accuracy.
f ( x ) = 0 with 4 decimal places of
• Use a
1
= 0 .
1 and b
1
= 0 .
3 to start. Record the results of each iteration.
• How many iterations did you have to do?
11.
(15 points) Use Newton’s method to approximate the solution to accuracy.
f ( x ) = 0 with 4 decimal places of
• Use x
1
= 0 .
5 as your initial guess. Record the results of each iteration.
• How many iterations did you have to do?
12.
(2 points) What is the solution to 2 x − cos (4 x ) = 0 up to 4 decimal places?
13.
(3 points) Show that the equation f ( x ) = 0 is equivalent to x = x − [2 x − cos (4 x )] . Also show that x = x − [2 x − cos (4 x )] is equivalent to x = x +cos (4 x )
3
.
14.
(12 points) Following the steps below, use the fixed-point algorithm to approximately solve f ( x ) = 0 .
• Using x
1
= 0 .
26 as your initial guess implement the algorithm for
Record the result of each iteration.
10 steps to solve x = x +cos (4 x )
3
.
• Compare your final iterate to your answer from the other two methods. How many decimal places of accuracy did you achieve with these 10 iterations?
15.
(3 points) Of the three methods you used, which performed best? Which performed worst?
16.
(15 points) Apply Newton’s method to the same equation, this time using x
1
= 1 as your initial guess.
• Implement the method for 10 steps, and record the result of each iteration.
• What do you notice about the output of each successive step?
• Why do you think this happens? Draw a picture of this circumstance along with the appropriate tangent lines, as we saw in class.